I did a biology undergraduate degree and often did reports where would statistically analyse our results. P value of less than 0.05 shows that the results are statistically significant. How do these tests actually know the data is significant? For example we might look at correlation and get a significant positive correlation between two variables. Given that variables can be literally anything in question, how does doing a few statistical calculations determine it is significant? I always thought there must be more nuance as the actual variables can be so many different things. It might show me a significant relationship for two sociological variables and also for two mathematical, when those variables are so different?
In: Mathematics
Generally, a statistical test can show that the probability that something happens under a certain assumption is very low (often, lower than 0.05 or 5% is used as a threshold). This generally gives evidence that the assumption is wrong. For example, I can flip a coin and I can make the assumption that the coin is fair (heads and tails are equally likely). If I flip it and it comes up heads 19 times out of 20, I can reason “if this coin were fair, then the chance of getting 19 heads is extremely small, so this coin is probably not fair”.
We can do the same thing for correlation. If we have a bunch of observations and each observation contains two variables (say, we measure the height and weight of a bunch of people), we can make the “assumption” that height and weight are not related to each other, aka, independent. Under this assumption, it is highly likely that the correlation coefficient will be quite close to 0. If we compute the correlation and find that it’s something much larger than 0, we know it is very unlikely for a bunch of unrelated random numbers to have a big correlation (by pure chance) so we conclude that our assumption was probably wrong. The statistical test gives us a way of quantifying exactly how unlikely this was.
If we change the variables from height and weight to intelligence and parents’ income, the math doesn’t change. We still make the assumption that intelligence and parental income are unrelated and then see how correlated intelligence and parents income are. If they were independent if each other, the correlation would probably be close to 0, and and a correlation far from 0 is very unlikely. The likelihood that independent random numbers end up correlated doesn’t depend on if the variables are biological, sociological, physical, etc.
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